Computing Hadron Properties - uni-graz.atcbl/teaching/MPI_2006.pdfRecent lattice QCD results on nucleon structure Search for Gluonic Excitations in Light Unconventional Mesons Towards

Computing Hadron PropertiesC.B. Lang :

MPI Munich, 17.10.2006

Computing Hadron Properties

C. B. LangInst. F. Physik / FB Theoretische Physik

Karl-Franzens-Universität Graz

C.B. Lang : Computing Hadron Properties

Concepts Methods The fermion adventure Applications: Hadron masses

1. Concepts

2. Methods

3. The fermion adventure

4. Applications: Hadron masses



Concepts of Quantumchromodynamics

QCD is the quantum field theory of quarks and gluons, defined by a Lagrangian (action)

This theory can be solved from first principles, withminimal number of input parameters (bare quark massesand a scale fixing parameter)Hadron properties should be computable from QCD



High energies, hard scattering:perturbative expansion

Low energies, hadron structure: non-perturbative methods

Mass, decay constants,matrix elements, distribution functions,form factors, finite temperature and density, …..



Quantization

(Euclidean space-time and )



Fundamental challenge

Non-perturbative resultsare hard to get.

Rigorous proof of confinement brings1,000.000 $ prize



QCD on a space-time grid: Lattice QCD

Gauge field

QuarksAnti-quarks

Ken Wilson

Lattice QCD is an approximation – not a modelIt opens a way to compute the path integral



Let‘s compute

…functional integral is approximated bycomputer (Monte Carlo) sampling of fieldconfigurations……..

Mike Creutz



Computational requirements

20 x 20 x 20 x 40 grid has- 11 M complex numbers for the gauge variables- 4 M complex numbers for quark variables

The Dirac operator beomes a huge matrix- e.g. 4M x 4M

Algorithms:MC simulationMatrix inversion by CGR type methods

Competitive projects need >1 TeraFlopYear



„Berlin Wall“

C. Urbach, LAT06



Custom computersPC Clusters„Self-built“ computers(apeNEXT, QCD-on-a-Chip)

BlueGeneL at Jülich (46 TFl, 8K nodes)

Altix at LRZ (26 TFl, 4K nodes )

QCDOC

QCDOC at BNL(20 TFl, 14 K nodes)



1. Concepts

2. Methods


4. Applications: Hadrons



HowTo: Simulate gauge fields nonperturbatively

The vacuum is not perturbativeGluons are represented by SU(3) matrices – like a 4D system of bosonicspinsMonte Carlo importance sampling of gauge configurationsMeasure all observables (propagators, matrix elements) in this gauge fieldbackground:„quenched approximation“

(Gluonic vaccuum fluctuations, movie © Leinweber et al.)



HowTo: Work with valence quarks

Sea quarksValence quarks



HowTo: Work with valence quarks

„quenched approximation“Valence quarks



HowTo: Measure masses

Compute the correlationfunction for an operator withcorrect quantum numbers, e.g.

GMOR

Mπ=280 MeV

BGR, Nucl.Phys. B677 (2004)



HowTo: Fix the scale

Simulate at some values of bare lattice couplings (g, mf)

All lattice quantities are dimensionless combinations(like e.g. mlatt= a mphys)

Trade one measured quantity with one experimental value to get the lattice spacing a

E.g. Sommer parameter = distance r0, where string tension givesmeasured Regge slope (r0

2 F(r0)=1.65)E.g. compute the (dimensionless lattice) pion decay constant(a fπ) and use fπ = 93 MeV Typical values used are a=0.05…0.25 fm



HowTo: Simulate dynamical quarks

$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Fermions(Grassmann)

Bosons„pseudo-fermions“

„Hybrid Monte Carlo“ = Molecular dynamics + MC step



HowTo remove the constraints?

Technical constraints

Conceptual issuesExcited states (and decays)?Chemical potential?Chiral theories (Standard model)?Supersymmetry?

Lattice spacinga 0

„continuum limit“Renormalization group, improvement (e.g. Symanzik)

Volume sizeL ∞

„thermodynamic limit“Finite size corrections, Chiral Perturbation Theory

Quark massm 0

…or at least small enough, i.e. mπ 0 ,„Chiral limit“, ChPT



Typical parameters for today‘s simulations

a=0.2 - 0.07 fm

Lattice size 123x24…323x64Spatial volume (1.5-2.5 fm)3

mπ around 350-800 MeV



Some topics (LAT06)

Algorithms for dynamical overlap fermionsLight dynamical fermions on the lattice: toward the chiral regime of QCDRooted staggered fermions: good, bad, or ugly?Status report on ILDG activitiesThe one-flavor quark condensate and related problemsThe rational hybrid Monte Carlo algorithmTwisted mass QCD for weak matrix elements

Algorithms

Heavy flavor physics from lattice QCDProgress in Kaon Physics on the LatticePushing Lattice to the Edge: Flavor PhysicsRecent lattice QCD results on nucleon structureSearch for Gluonic Excitations in Light Unconventional MesonsTowards a quantitative understanding of the Delta I=1/2 ruleTwisted mass QCD for weak matrix elements

Hadron properties

Color superconductivity in ultra-dense quark matterHot QCD and warm dark matterLattice QCD at finite densityQCD phase diagram: an overviewRecent progress in finite temperature lattice QCDStrong Correlations in Hot QCDThe sign problem in the epsilon-regime of QCD

Extreme environments

Advances and applications of lattice supersymmetryD-branes and coherent topological charge structure in QCDLattice Field Theory Methods in Modern BiophysicsThe search for the fundamental QCD string in anti de Sitter spaceQCD physics beyond the standard model and LHC searches

„Transgressing boundaries“



1. Concepts

2. Methods





A problem for lattice QCD: chiral symmetry

Continuum QCD: Chiral symmetry SU(2)R x SU(2)L is spontaneouslybroken: SU(2) multiplets + Goldstone pions

The theory should allow for explicit chiral symmetry, such that it canbe broken spontaneously!

Local term

Lattice QCD:

This „Ginsparg-Wilson condition“ is violated bysimple Dirac operators (simple fermion actions)



What fermions are on the market?

StaggeredWilson-improvedTwisted mass

Domain WallFixed PointChirally Improved

Overlap

Non-GW-type

GW-type

GW-exact



Non-GW actions: „Clash of discretisations“

Staggered (improved: Asqtad)

Wilson (improved: clover)Twisted mass



Non-GW actions: Staggered type

Sugar (LAT06)

Staggered (improved: Asqtad, MILC collaboration)

Quark degrees distributed in hypercubeRemnant chiral symmetrySimple to implementHarder to interpret (taste splitting)4 fermions:

- 4th root trick: (det D)1/4 ???



Non-GW actions: Wilson type

m m ?

lGW lWilsonWilson (improved: clover)Simple to implementDoubler modesProblem with small quarkmasses: Spurious low-lyingeigenmodes of the Dirac operator

Twisted mass (© Frezotti,Grassi, Sint, Weisz)

Wilson + „twist“ i mg5t3

No spurious modesBreaks parity and flavor

DelDebbio et al. JHEP 0602(2006)



GW-type fermions…fewer or no spurious modes…can reach smaller pion and quark masses

Domain Wall5th „dummy“ dimension (© Kaplan, Furman, Shamir)

Fixed pointRG transformation, parametrisation (© Hasenfratz, Niedermeier)

CI (chirally improved)Systematic expansion, parameterisation (© Gattringer, Hip, L.)

Overlap„exact“ but very expensive (© Neuberger)

1+γ5 sign(γ5 DW) m

lGW

+ . . .+ += +



1. Concepts

2. Methods





Fundamental constants

2004 MILC/HPQCD/UKQCD/Fermilab „Gold plated“(?) results

2006 values range

CP-PACS, JLQCD- mud= 3.5 (4) MeV- ms = 92 (9) MeV- Fπ = 141(9) MeV- fK = 161(9) MeV

Davies et al. PRL92 (04)022001

MILC- mu = 2.0 (3) MeV, md =4.6 (5) MeV- ms = 90 (9) MeV- fπ = 129 (3) MeV - fK = 155 (3) MeV



Ground state masses

CP-PACS (2002, Wilson, quenched, extrapolated)

MILC(2004, staggered, dynamical, extrapolated)



How to identify excited states?

Propagator = sum of exponential decay terms:

ground state (large t)

excited states (smaller t)

Previous attempts: biased estimators (Bayesian analysis), maximum entropy,...

Significant improvement: Variational analysis

Mathur et al. (05), Lee et al (03),Sasaki et al. (05), Juge et al. (06),Zanotti et al. (03), Melnichouk et al.(03)

Burch et al. (03-06), Basak et al. (05, 06)



Variational analysis (Michael, Lüscher/Wolff)

Use several interpolators

Compute all cross-correlations

Solve the generalized eigenvalue problem:

The eigenvectors are „fingerprints“ of the state

Allows to separate ghost contributions (cf. Burch et al.)



Analysis of excited hadron masses

Interpolation fields:

E.g. for nucleon

Allow for different quark sourceshapes

Analyse cross-correlations



Exciting nucleons

Roper Level crossing (from + - + - to + + - -)?

0.00 0.40 0.80 1.20

(mπ)2 [GeV]

2

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

MN

[G

eV]

λ(1), a = 0.148fm

λ(2), a = 0.148fm

λ(1), a = 0.119fm

λ(2), a = 0.119fm

λ(3), a = 0.119fm

negative parity

N(1535)

N(1650)

N(2090)

0.00 0.40 0.80 1.20

(mπ)2 [GeV]

2

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

MN

[G

eV]

λ(1), a = 0.148fm

λ(2), a = 0.148fm

λ(3), a = 0.148fm

λ(1), a = 0.119fm

λ(2), a = 0.119fm

λ(3), a = 0.119fm

positive parity

N(938)

N(1440)

N(1710)

Burch et al., 2006



Exciting nucleons

Roper Level crossing (from + - + - to + + - -)?

0.00 0.40 0.80 1.20

(mπ)2 [GeV]

2

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

MN

[G

eV]

λ(1), a = 0.148fm

λ(2), a = 0.148fm

λ(1), a = 0.119fm

λ(2), a = 0.119fm

λ(3), a = 0.119fm

negative parity

N(1535)

N(1650)

N(2090)

0.00 0.40 0.80 1.20

(mπ)2 [GeV]

2

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

MN

[G

eV]

λ(1), a = 0.148fm

λ(2), a = 0.148fm

λ(3), a = 0.148fm

λ(1), a = 0.119fm

λ(2), a = 0.119fm

λ(3), a = 0.119fm

positive parity

N(938)

N(1440)

N(1710)

Burch et al., 2006

?



Chiral PT: Expansion around Mπ = 0

Chiral dynamics:Goldstone pion

- Effective theory for vanishing pion masses

- Quark masses as perturbation- Low energy theorems like PCAC are

exact

Systematic expansion in Mπ/Lch , E/Lch :- Lagrangian with phenomenological

parameters

(Weinberg, Leutwyler, …):

Meissner (LAT05)



Chiral perturbation theory for excited states?

… new scale parameters(non-vanishing in the chiral limit) … new assumptions…

Roperbehaves similar to theNucleon ?

Bernard, Hemmert, Meissner (2004)



Baryons (chiral extrapolations)

1.0

1.5

2.0

2.5

MB [

GeV

]

N Σ Ξ Λ ∆++ Ω− −

negative parity

1.0

1.5

2.0

2.5

MB [

GeV

]

N Σ Ξ Λ ∆++ Ω− −

positive parity

Burch et al., 2006



Baryons (chiral extrapolations)

1.0

1.5

2.0

2.5

MB [

GeV

]

N Σ Ξ Λ ∆++ Ω− −

negative parity

Predictions:W 1st excited state, pos.parity: 2300(70) MeVW ground state, neg.parity: 1970(90) MeVX ground state, neg.parity: 1780(90) MeVX 1st excited stated, neg.parity: 1780(110) MeV

1.0

1.5

2.0

2.5

MB [

GeV

]

N Σ Ξ Λ ∆++ Ω− −

positive parity

Burch et al., PRD 74 (2006)



What to expect for the next years?

Dynamical quarks („full QCD“, 2+1 flavors)

2008: reasonable dynamical results for mp around 200 MeV 2010: reliable dynamical results for mp around 200 MeV 2012: reasonable dynamical results for mp around 140 MeV 2014: reliable dynamical results for mp around 140 MeV



The future

Prediction is very difficult, especially about the future(US: Yogi Berra)



The future

Prediction is very difficult, especially about the future(EU: Niels Bohr)



Thank you !

Documents

Computing Hadron Properties - uni-graz.atcbl/teaching/MPI_2006.pdfRecent lattice QCD results on nucleon structure Search for Gluonic Excitations in Light Unconventional Mesons Towards